The HERA accelerator, an electron-proton collider facility based at the DESY laboratory in Hamburg, started delivering luminosity to the ZEUS experiment in May 1992. The short time of 96ns between bunch crossings and the fact that the primary source of detector activity in ZEUS does not come from the e-p interactions of interest but rather from other sources makes a high performance three level trigger system essential to data-taking. We present one part of the ZEUS trigger system, the Central Tracking Detector First Level Trigger (CTD-FLT). The principle of operation and the hardware implementation are discussed at length. A prototype system which was used during the early stages of ZEUS data-taking is described. We detail the principles of the CTD-FLT hardware testing and discuss aspects of the First Level Trigger data from the 1993 running period. During this time the CTD trigger has proved to be an invaluable part of the overall system. The high energies accessible at HERA make it possible to probe a variety of new physics models. The second part of this thesis focuses on one such model - Supersymmetry (SUSY). After an introduction to the model, a search for R-parity violating SUSY in the ZEUS data from the 1992 running period is described. No evidence for such signals has been found and hence we set upper limits on the mass scale of the theory. Finally we present a Monte Carlo study of the potential discovery reach of HERA for R-parity violating SUSY models, and come to the conclusion that HERA is an ideal place to look for such phenomena.